The (10, 3)-a Network

Above is a sculptural concept based on
the crystal structure (10,3)-a,
described
in the 1950's by the chemist A.F. Wells. I have taken that
arrangement of vertices and edges and wrapped a smooth organic volume
around it using a general-purpose algorithm described here. The result is a
"slimmed down" gyroid surface. Above is a 4-by-4-by-4
block of the lattice structure made as a 3.5 inch nylon model using
selective laser sintering. Check out the various views below, but it is
impossible to convey its 3D structure with 2D
images. It would be better to look at the stl
file in a 3D
viewer or use it to make your own physical copy.

The above rendering only hints
at how cool the real physical model is when you hold it.
For one thing, my version stands up on one corner, balanced nicely on
the three feet in the
lower-left.

Looking down the X, Y, or Z axis
drections, you see the above pattern of large and small tunnels.

The above rendering gives more perspective.

Looking down a 3-fold axis you see a hexagonal pattern of tunnels.

In any 110 direction, you get this pattern.

In other directions, you get other patterns. It is a very rich
object!

Wells points out that the structure
can
interpenetrate space with its mirror image.
This colored rendering clarifies that there are two separate components.

A physical model of the two interlocked structures is immensely complex.
The two parts in the above 3.5 inch nylon model are free to jiggle
slightly.

Here is the view of the doubled structure along an X, Y, or Z direction.
This one is a 5-by-5-by-5 block of doubly covered unit cells.
(This doubled model also stands up nicely, balanced on a corner!)

The underlying structure of
vertices and edges can be conveniently made with Zometool.
The (10, 3)-a network is just the green edges in the above unit cell.
The
yellow lines are its 3-fold axes.

The structure is described in A.F.
Wells' 1956 book The Third Dimension
in Chemistry.
It was recently
rediscovered by Toshikazu Sunada
and called "K4" in this AMS
Notices paper.
His subsequent letter updating that paper is online here
and an update by Alan Schoen is here.
Also see H.S.M. Coxeter's "On Laves'
Graph of Girth Ten" Canadian Journal of Math, 1955.
The first presentation of the structure was apparently in 1933, in this
German publication:
H. Heesch, F. Laves, "Über dünne kugelpackungen," Z. Kristallogr., 1933, 85, 443-453.
A history with additional references is: Stephen T. Hyde, Michael
O'Keeffe, Davide M. Proserpio,
"A short history of an elusive yet ubiquitous structure in chemistry,
materials and mathematics," Angewandte Chemie, International
Edition, 2008, 47 (42): 7996-8009. (abstract)

It leads to the above Heesch-Laves loose-packing of spheres.

Double Diamond

On a related topic, above is the analogous idea of the "double diamond
lattice".

This is a small (1.6 inch) physical SLS model. It consists of two
copies of the structure below.
They are free to wiggle slightly relative to each other. It
is said to occur in a sodium-thallium alloy,
where the atoms of each metal take the positions of one of the
interlinked
carbon lattices [reference].